The action of the quantum mechanical volume operator, introduced in connection with a symmetric representation of the three-body problem and recently recognized to play a fundamental role in discretized quantum gravity models, can be given as a second-order difference equation which, by a complex phase change, we turn into a discrete SchrÃ¶dinger-like equation. The introduction of discrete potential-like functions reveals the surprising crucial role here of hidden symmetries, first discovered by Regge for the quantum mechanical 6j symbols; insight is provided into the underlying geometric features. The spectrum and wavefunctions of the volume operator are discussed from the viewpoint of the Hamiltonian evolution of an elementary ‘quantum of space’, and a transparent asymptotic picture of the semiclassical and classical regimes emerges. The definition of coordinates adapted to the Regge symmetry is exploited for the construction of a novel set of discrete orthogonal polynomials, characterizing the oscillatory components of torsion-like modes.

Publication

Journal of Physics A: Mathematical and Theoretica, **2013**, *46*, 175303

Date

April, 2013

Bibtex:

```
@Article{Aquilanti:2013qx,
Title = {{Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials}},
Author = {Aquilanti, Vincenzo and Marinelli, Dimitri and Marzuoli, Annalisa},
Journal = {Journal of Physics A: Mathematical and Theoretical},
Year = {2013},
Pages = {175303},
Volume = {46},
Archiveprefix = {arXiv},
Doi = {10.1088/1751-8113/46/17/175303},
Eprint = {1301.1949},
Primaryclass = {quant-ph},
}
```